An Introduction to the Display Calculus

An introduction to the display calculus

Revantha Ramanayake

Technical University of Vienna (TU Wien)

Ninth International Tbilisi Summer School in Logic and Language Sep. 30–Oct. 4, 2013

Revantha Ramanayake (TU Wien) An introduction to the display calculus 1 / 66 A (very) brief history of proof theory

The so-called foundational crisis of mathematics (early 1900s) arose from various challenges to the implicit assumption that consistency of the ‘foundations of mathematics’ could be shown within mathematics For example, the discovery of Russell’s paradox (1902) showed that naive set theory was an inadequate foundation. The need for a precise development of the underlying logical systems became apparent. The intention is to present a narrative placing the display calculus in a broader context. Introduction not intended to be at all comprehensive!

Revantha Ramanayake (TU Wien) An introduction to the display calculus 2 / 66 Hilbert’s Program (around 1922)

Proofs are the essence of mathematics — to establish a theorem.. present a proof! Historically, proofs were not the objects of mathematical investigations (unlike numbers, triangles. . . ) In Hilbert’s Proof theory: proofs are mathematical objects.

Hilbert’s Program: 1 Formalise the whole of mathematical reasoning in a formal theory T 2 Prove the consistency of T by ‘ﬁnitistic’ means

T is consistent if there is no formula A such that A ∧ ¬A is derivable in T .

Revantha Ramanayake (TU Wien) An introduction to the display calculus 3 / 66 Hilbert calculus

Mathematical investigation of proofs f formally deﬁnition of proof Hilbert calculus fulﬁls this role.

A Hilbert calculus for propositional classical logic. Axiom schemata:

Ax 1: A → (B → A) Ax 2: (A → (B → C)) → ((A → B) → (A → C)) Ax 3: (¬A → ¬B) → ((¬A → B) → A) and the rule of modus ponens:

AA → B B Read A ↔ B as (A → B) ∧ (B → A). More axioms:

Ax 4: A ∨ B ↔ (¬A → B) Ax 5: A ∧ B ↔ ¬(A → ¬B)

Revantha Ramanayake (TU Wien) An introduction to the display calculus 4 / 66 Derivation of A → A

Deﬁnition

A formal proof (derivation) of B is the ﬁnite sequence C1, C2,..., Cn ≡ B of formulae where each element Cj is an axiom instance or follows from two earlier elements by modus ponens.

1 ((A → ((A → A) → A)) → ((A → (A → A)) → (A → A))) Ax 2 2 (A → ((A → A) → A)) Ax 1 3 ((A → (A → A)) → (A → A)) MP: 1 and 2 4 (A → (A → A)) Ax 1 5 A → A MP: 3 and 4

Not easy to ﬁnd! Proof has no clear structure (wrt A → A)

Revantha Ramanayake (TU Wien) An introduction to the display calculus 5 / 66 Gödel’s second incompleteness theorem (1931)

Theorem Let T be a consistent theory containing arithmetic. Then there is no proof of consistency of T in T (ie. T 6` Con(T )).

Destroys Hilbert’s program (assuming finitistic reasoning can be formalised in arithmetic, as was believed) If it is the case that (i) consistency of mathematics can be shown by finitistic reasoning, and (ii) arithmetic can formalise finitistic reasoning Then arithmetic can show the consistency of mathematics (and hence arithmetic). Contradiction. So if we believe that (ii) holds, then (i) cannot hold.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 6 / 66 Natural deduction and the sequent calculus

Gentzen: proving consistency of arithmetic in weak extensions of ﬁnitistic reasoning. Hilbert calculus not convenient for studying the proofs (lack of structure). Gentzen introduces Natural deduction which formalises the way mathematicians reason. Gentzen introduced a proof-formalism with even more structure: the sequent calculus. Sequent calculus built from sequents X ` Y where X, Y are lists/sets/multisets of formulae

Revantha Ramanayake (TU Wien) An introduction to the display calculus 7 / 66 Sequent calculus

antecedent succedent z }| { z }| { sequent: A , A ,..., A ` B , B ,..., B 1 2 m |{z} 1 2 n turnstile

sequent calculus rule: k ≥ 0 premises z }| { (S0, S1,..., Sk S1 ... Sk are sequents) S |{z} conclusion

Typically a rule for introducing each connective in the antecedent and succedent. A 0-premise rule is called an initial sequent

Deﬁnition (derivation) A derivation in the sequent calculus is an initial sequent or a rule applied to derivations of the premise(s).

Revantha Ramanayake (TU Wien) An introduction to the display calculus 8 / 66 The sequent calculus SCp for classical logic Cp

init ⊥l p, X ` Y , p ⊥, X ` Y X ` Y , A A, X ` Y ¬l ¬r ¬A, X ` Y X ` Y , ¬A A, B, X ` Y X ` Y , AX ` Y , B ∧l ∧r A ∧ B, X ` Y X ` Y , A ∧ B A, X ` YB, X ` Y X ` Y , A, B ∨l ∨r A ∨ B, X ` Y X ` Y , A ∨ B X ` Y , AB, X ` Y A, X ` Y , B →l →r A → B, X ` Y X ` Y , A → B

Here X, Y are sets of formulae (possibly empty) Aside: differs from the calculus Gentzen used (not important) Gödel’s incompleteness theorem does not apply since this logic does not contain arithmetic

Revantha Ramanayake (TU Wien) An introduction to the display calculus 9 / 66 Soundness and completeness of SCp for Cp

Need to prove that SCp is actually a sequent calculus for Cp.

Theorem For every formula A we have: `A is derivable in SCp ⇔ A ∈ Cp.

(⇒) direction is soundness. (⇐) direction is completeness.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 10 / 66 Proof of completeness

Need to show: A ∈ Cp ⇒ `A derivable in SCp. First show that A, X ` Y , A is derivable (exercise). Show that every axiom of Cp is derivable (easy, below) and modus ponens can be simulated in SCp (not easy)

B, A ` C, B r, B, A ` C B, A ` C, A B → C, B, A ` C A, A → (B → C) ` C, A B, A, A → (B → C) ` C A, A → B, (A → (B → C)) ` C A → B, (A → (B → C)) ` (A → C) (A → (B → C)) ` (A → B) → (A → C) ` (A → (B → C)) → ((A → B) → (A → C))

Revantha Ramanayake (TU Wien) An introduction to the display calculus 11 / 66 How to simulate modus ponens

Gentzen’s solution: to simulate modus ponens (below left) ﬁrst add a new rule (below right) to SCp: AA → B X ` Y , AA, X ` Y cut B X ` Y The following instance of the cut-rule illustrates the simulation of modus ponens.

A ` AB ` B ` A → B A → B, A ` B ` A A ` B cut ` B So: A ∈ Cp ⇒ `A derivable in SCp + cut!

Revantha Ramanayake (TU Wien) An introduction to the display calculus 12 / 66 Proof of soundness

Need to show: `A derivable in SCp + cut ⇒ A ∈ SCp. We need to interpret SCp + cut derivations in Cp.

For sequent SA1, A2,..., Am ` B1, B2,..., Bn

deﬁne translation τ(S) A1 ∧ A2 ∧ ... ∧ Am → B1 ∨ B2 ∨ ... ∨ Bn

Comma on the left is conjunction, comma on the right is disjunction. Translations of the intial sequents are theorems of Cp

p ∧ X → Y ∨ p ⊥ ∧ X → Y

Show for each remaining rule ρ: if the translation of every premise is a theorem of Cp then so is the translation of the conclusion.

A, X ` B A ∧ X → B For need to show: X ` A → B X → (A → B)

Revantha Ramanayake (TU Wien) An introduction to the display calculus 13 / 66 The cut-rule is undesirable in SCp + cut

We have shown Theorem For every formula A we have: `A is derivable in SCp + cut ⇔ A ∈ Cp.

The subformula property states that every formua in a premise appears as a subformula of the conclusion. If all the rules of the calculus satisfy this property, the calculus is analytic Analyticity is crucial to using the calculus (for consistency, decidability. . . ) SCp + cut is not analytic because: X ` Y , AA, X ` Y cut X ` Y We want to show: `A is derivable in SCp ⇔ A ∈ Cp

Revantha Ramanayake (TU Wien) An introduction to the display calculus 14 / 66 Gentzen’s Hauptsatz (main theorem): cut-elimination

Theorem Suppose that δ is a derivation of X ` Y in SCp + cut. Then there is a transformation to eliminate instances of the cut-rule from δ to obtain a derivation δ0 of X ` Y in SCp.

Since `A is derivable in SCp + cut ⇔ A ∈ Cp:

Theorem For every formula A we have: ` A is derivable in SCp if and only if A ∈ Cp.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 15 / 66 Applications: Consistency of classical logic

Consistency of classical logic is the statement that A ∧ ¬A < Cp. Theorem Classical logic is consistent.

Proof by contradiction. Suppose that A ∧ ¬A ∈ Cp. Then A ∧ ¬A is derivable in SCp (completeness). Let us try to derive it (read upwards from ` A ∧ ¬A):

A ` ` A ` ¬A ` A ∧ ¬A So ` A and A ` are derivable. Thus ` must be derivable in SCp + cut (use cut) and hence in SCp (by cut-elimination). This is impossible (why?) QED.

Theorem Decidability of Cp.

Given a formula A, do backward proof search in SCp on ` A. Since termination is guaranteed, we can decide if A is a theorem or not. QED.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 16 / 66 Looking beyond the sequent calculus

Aside from proofs of consistency, proof-theoretic methods enable us to extract information from the proofs and about the logic (a fact already recognised by Gentzen). Many more logics of interest than just ﬁrst-order classical and intuitionistic logic How to give a proof-theory to these logics? Want analytic calculi with modularity In a modular calculus we can add rules corresponding to (suitable) axiomatic extensions and preserve analyticity.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 17 / 66 Some nonclassical logics

Modal logics extend classical language with modalities and ^. The modalities were traditionally used to qualify statements like “it is possible that it will rain today". Tense logics include the temporal modalities and . Closed under modus ponens and necessitation rule (A/A).

An intermediate logic L is a set of formulae closed under modus ponens such that intuitionistic logic Ip ⊆ L ⊆ Cp.

Starting with the sequent calculus SCp, if we consider a sequent X ` Y to be built from lists X, Y (rather than sets or multisets) then A, A, X ` Y and A, X ` Y are no longer identical (no contraction). Also A, B, X ` Y and B, A, X ` Y are not identical (no exchange). The logics obtained by removing these properties are called substructural logics.

Sequent calculus inadequate for treating these logics (eg. no analytic sequent calculus for S5 despite analytic sequent calculus for S4)

Revantha Ramanayake (TU Wien) An introduction to the display calculus 18 / 66 Display Calculus

Introduced as Display Logic (Belnap, 1982). Extends sequent calculus by introducing new structural connectives that interpret the logical connectives (enrich language) A structure is built from structural connectives and formulae. A display sequent: X ` Y for structures X and Y Display property. A substructure in X[U] ` Y equi-derivable (displayable) as U ` W or W ` U for some W . Key result. Belnap’s general cut-elimination theorem applies when the rules of the calculus satisfy C1–C8 (display conditions) Display calculi have been presented for substructural logics, modal and poly-modal logics, tense logic, bunched logics, bi-intuitionistic logic. . .

Revantha Ramanayake (TU Wien) An introduction to the display calculus 19 / 66 Display calculi generalise the sequent calculus

Here is the sequent calculus SCp once more: init ⊥l p, X ` Y , p ⊥, X ` Y X ` Y , A A, X ` Y ¬l ¬r ¬A, X ` Y X ` Y , ¬A A, B, X ` Y X ` Y , AX ` Y , B ∧l ∧r A ∧ B, X ` Y X ` Y , A ∧ B A, X ` YB, X ` Y X ` Y , A, B ∨l ∨r A ∨ B, X ` Y X ` Y , A ∨ B X ` Y , AB, X ` Y A, X ` Y , B →l →r A → B, X ` Y X ` Y , A → B

Revantha Ramanayake (TU Wien) An introduction to the display calculus 20 / 66 Display calculi generalise the sequent calculus

Let’s add a new structural connective ∗ for negation. init ⊥l p, X ` Y , p ⊥, X ` Y ∗A, X ` Y X ` Y , ∗A ¬l ¬r ¬A, X ` Y X ` Y , ¬A A, B, X ` Y X ` Y , AX ` Y , B ∧l ∧r A ∧ B, X ` Y X ` Y , A ∧ B A, X ` YB, X ` Y X ` Y , A, B ∨l ∨r A ∨ B, X ` Y X ` Y , A ∨ B X ` Y , AB, X ` Y A, X ` Y , B →l →r A → B, X ` Y X ` Y , A → B

Revantha Ramanayake (TU Wien) An introduction to the display calculus 21 / 66 Add the display rules

The addition of the following rules permit the display property:

Deﬁnition (display property) The calculus has the display property if for any sequent X ` Y containing a substructure U, there is a sequent U ` W or W ` U for some W such that X ` Y X ` Y or U ` W W ` U We say that U is displayed in the lower sequent.

X, Y ` Z X, Y ` Z X ` Y , Z X ` Z, ∗Y Y ` ∗X, Z X, ∗Z ` Y X ` Y , Z ∗X ` Y X ` ∗Y ∗Y , X ` Z ∗Y ` X Y ` ∗X ∗ ∗ X ` Y X ` ∗ ∗ Y X ` •Y X ` Y X ` Y •X ` Y

Revantha Ramanayake (TU Wien) An introduction to the display calculus 22 / 66 Using the display rules

Examples: ∗(A, ∗B) ` ∗(C, D) ∗(A, ∗B) ` ∗(C, D) ∗ ∗ (C, D) ` A, ∗B C, D ` ∗ ∗ (A, ∗B) ∗A, ∗ ∗ (C, D) ` ∗B D ` ∗C, ∗ ∗ (A, ∗B) B ` ∗(∗A, ∗ ∗ (C, D)) D is displayed B is displayed Exercise. Prove that the display property holds for this calculus. Also see Kracht, 1996.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 23 / 66 Specify the properties of the structural connectives

We want weakening, contraction, exchange, associativity. Here I is a structural constant for the empty list.

X ` Z X ` Z I ` Y I, X ` Z X ` I, Z ∗I ` Y X ` I X ` Z X ` Z X ` ∗I Y , X ` Z X, Y ` Z X, Y ` Z Z ` X, Y X, X ` Z Y , X ` Z Z ` Y , X X ` Z

Z ` X, X X1, (X2, X3) ` Z Z ` X1, (X2, X3) ` Z X (X1, X2), X3 ` Z Z ` (X1, X2), X3

Revantha Ramanayake (TU Wien) An introduction to the display calculus 24 / 66 Display calculi generalise the sequent calculus

The presence of the display rules permit the following rewriting of the rules: ⊥l p ` p init ⊥` I ∗A ` Y X ` ∗A ¬l ¬r ¬A ` Y X `¬ A A, B ` Y X ` AX ` B ∧l ∧r A ∧ B ` Y X ` A ∧ B A ` YB ` Y X ` A, B ∨l ∨r A ∨ B ` Y X ` A ∨ B X ` AB ` Y A, X ` B →l →r A → B ` ∗X, Y X ` A → B The formulae are called principal formulae. The X, Y are context variables.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 25 / 66 Sequent calculus to display calculus

From a procedural point of view, we obtained the display calculus δCp for Cp from the sequent calculus by 1 Addition of a structural connective ∗ for negation 2 Addition of the display rules to yield the display property 3 Additional structural rules for exchange, weakening, contraction etc. 4 Rewriting the logical rules so the principal formulae in the conclusion are all of the antecedent or succedent

We will consider a theoretical viewpoint shortly.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 26 / 66 The display calculus δKt for tense logic Kt

Tense logics extend the classical language with the modal operators ^, and the tense operators , .

and ^ are duals. Similarly and . Ie. axioms:

A ↔ ¬^¬A A ↔ ¬¬A

The following ‘residuation’ property holds in the basic tense logic Kt.

A → B if and only if A → B

Identifying residuation is crucial for constructing a display calculus (more later).

Revantha Ramanayake (TU Wien) An introduction to the display calculus 27 / 66 Display rules from residuation

We saw that: A → B ⇔ A → B

Introduce a new structural connective • for (, ) (ie. in the antecedent, in the succedent). •X ` Y Add the display rules X ` •Y and additional structural rules (for necessitation)

` ` I Y (Ml) X I (Mr) •I ` Y X ` •I

Revantha Ramanayake (TU Wien) An introduction to the display calculus 28 / 66 A display calculus δKt for tense logic Kt

p ` p ⊥ ` I

∗A ` X X ` ∗A ¬l ¬r ¬A ` X X ` ¬A A ◦ B ` X X ` AX ` B ∧l ∧r A ∧ B ` X X ` A ∧ B A ` XB ` X X ` A ◦ B ∨l ∨r A ∨ B ` X X ` A ∨ B X ` AB ` Y A, X ` B →l →r A → B ` ∗X, Y X ` A → B A ` X X ` •A l r A ` •X X ` A ∗ • ∗A ` X X ` A ^l ^r ^A ` X ∗ • ∗X ` ^A •A ` X X ` A l r A ` X •X ` A A ` X X ` ∗ • ∗A l r A ` ∗ • ∗X X ` A

Revantha Ramanayake (TU Wien) An introduction to the display calculus 29 / 66 Display Property

Theorem. Every substructure Z appearing in X ` Y can be displayed as the whole of the antecedent (Z ` U) or the whole of the succedent (U ` Z) for suitable U.

X, Y ` Z X, Y ` Z X ` Y , Z X ` Z , ∗Y Y ` ∗X, Z X, ∗Z ` Y

X ` Y , Z ∗X ` Y X ` ∗Y ∗Y , X ` Z ∗Y ` X Y ` ∗X

∗ ∗ X ` Y X ` ∗ ∗ Y X ` •Y X ` Y X ` Y •X ` Y Example.

∗ • ∗p ` q p ` •(q, ∗ • ∗r) ∗q ` • ∗ p •p ` q, ∗ • ∗r • ∗ q ` ∗p ∗q, •p ` ∗ • ∗r p ` ∗ • ∗q ∗ • ∗(∗q, •p) ` r p is displayed r is displayed

Revantha Ramanayake (TU Wien) An introduction to the display calculus 30 / 66 Translating sequents into formulae

We have seen that a sequent X ` Y in δKt is built from structures X, Y :

Struc ::= tense formula | I | (X, X) | •X | ∗X

Deﬁne the translation functions l and r from structures into tense formulae:

l(A) = A r(A) = A l(I) = > r(I) = ⊥ l(∗X) = ¬r(X) r(∗X) = ¬l(X) l(X, Y ) = l(X) ∧ l(Y ) r(X, Y ) = r(X) ∨ r(Y ) l(•X) = l(X) r(•X) = r(X)

The sequent X ` Y is interpreted as l(X) → r(Y ).

Revantha Ramanayake (TU Wien) An introduction to the display calculus 31 / 66 Soundness of δKt for Kt

It sufﬁces to prove that if the translations of the premises are theorems of Kt, then so is the translation of the conclusion (this has to be done for all rules). Eg. consider the display rule (below left) and its formula translation (below right) •X ` Y l(X) → r(Y ) X ` •Y l(X) → r(Y ) We already noted that above right holds in Kt. Aside. Helpful to prove soundness with respect to frame semantics for Kt.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 32 / 66 Completeness of δKt for Kt

As in the case of the sequent calculus, we can prove completeness by deriving all the axioms, and A simulating modus ponens and necessitation A The following sufﬁces: X ` AA ` Y I ` Y cut (Ml) X ` Y •I ` Y The cut-rule applies only to a formula (and not a structure!) To obtain an analytic calculus we need to eliminate cut.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 33 / 66 Belnap’s general cut-elimination theorem

Belnap showed that any display calculus satisfying the display conditions has cut-elimination. The display conditions C1–C8 are syntactic conditions on the rules of the calculus. Theorem A display calculus that satisﬁes the Display Conditions C2–C8 has cut-elimination. If C1 is satisﬁed, then the calculus has the subformula property.

Proof ‘follows’ Gentzen’s cut-elimination, uses display property. Only C8 is non-trivial to verify.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 34 / 66 Display conditions

∗ • ∗A ` X X, Y ` Z A ` X (^l) (∗r) ^A ` X X ` Z, ∗Y A ` X

(C1) Each schematic formula variable occurring in a premise of a rule ρ , cut is a sub-formula of some schematic formula variable in the conclusion of ρ. (C2) A parameter is an occurrence of a schematic structure variable in the rule schema. Occurrences of the identical structure variable are said to be congruent to one another (really a deﬁnition) (C3) Each parameter is congruent to at most one structure variable in the conclusion. Ie. no two structure variables in the conclusion are congruent to each other.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 35 / 66 Display conditions

∗ • ∗A ` X X, Y ` Z X ` Y (^l) (∗r) ^A ` X X ` Z, ∗Y X, X ` Y

Revantha Ramanayake (TU Wien) An introduction to the display calculus 36 / 66 Display conditions

∗ • ∗A ` X X ◦ Y ` Z X ◦ Y ` Z (^l) (∗r) ^A ` X X ` Z ◦ ∗Y X ` Z ◦ Y

(C4) Congruent parameters are all either a-part or s-part structures. (C5) A schematic formula variable in the conclusion of an inference rule ρ is either the entire antecedent or the entire succedent. This formula is called a principal formula of ρ. (C6/7) Each inference rule is closed under simultaneous substitution of arbitrary structures for congruent parameters.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 37 / 66 Display conditions

∗ • ∗A ` X X ◦ Y ` Z A, B, X ` Y (^l) (∗r) ^A ` X X ` Z ◦ ∗Y A ∧ B, X ` Y

Revantha Ramanayake (TU Wien) An introduction to the display calculus 38 / 66 Display conditions

∗ • ∗A ` X X ◦ Y ` Z (^l) (∗r) ^A ` X X ` Z ◦ ∗Y

(C8) If there are inference rules ρ and σ with respective conclusions X ` A and A ` Y with formula A principal in both inferences (in the sense of C5) and if cut is applied to yield X ` Y , then X ` Y is identical to either X ` A or A ` Y ; or it is possible to pass from the premises of ρ and σ to X ` Y by means of inferences falling under cut where the cut-formula is always a proper sub-formula of A.

∗ • ∗A ` Y X ` A ∗ • ∗A ` Y drs ^r l X ` A A ` ∗ • ∗Y ∗ • ∗X ` A A ` Y ^ ^ ^ X ` ∗ • ∗Y ∗ • ∗X ` Y drs ∗ • ∗X ` Y

Revantha Ramanayake (TU Wien) An introduction to the display calculus 39 / 66 Display calculi for modal logics

A display calculus δK for the modal logic K can be obtained by deleting the introduction rules for the tense operators and . By cut-elimination and the conservativity of tense logic over modal logic, any derivation of a modal formula does not require the use of tense operators. However, using the display rules in δK may result in • in the antecedent. p ` p p → p p ` •p l p → p dr •p ` p p ` p Recall that cut-elimination in the display calculus relies on the display property.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 40 / 66 Kracht’s structural rule extensions of Kt

Deﬁnition A primitive tense axiom has the form A → B where both A and B are constructed from propositional variables and > using {∧, ∨, ^, } and A contains each propositional variable at most once.

Some examples of primitive tense axioms

^Â → Â Â → Â ^(A ∧ B) → ^(A ∨ ^B)

First two axioms are primitive tense equivalents of A → A (transitivity) and ^A → ^A (connectedness) respectively. Theorem (Kracht)

Let L be a tense logic. Then L is an axiomatic extension of Kt by primitive tense axioms iff there is a proper structural rule extension of δKt for L.

A proper structural rule is a structural rule satisfying C1–C8.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 41 / 66 Structural rules from primitive tense axioms

Let A → B be a primitive tense axiom. Since

^(C ∨ D) ↔ ^C ∨ ^D (C ∨ D) ↔ C ∨ D (C ∨ D) ∧ E ↔ (C ∧ E) ∨ (D ∧ E) W W So A → B ↔ ( i≤m Ci ) → j≤n Dj where Ci , Dj built from >, ∧, and ^. Then A → B is equivalent to the following axioms: _ _ C1 → Dj ... Cm → Dj j≤n j≤n

Revantha Ramanayake (TU Wien) An introduction to the display calculus 42 / 66 Structural rules from primitive tense axioms (ctd)

W Now Cj → j≤n Dj is equivalent to the rule

σ(D1) ` Y . . . σ(Dn) ` Y ρi σ(Cj ) ` Y where

σ(>) = I

σ(p) = Xp σ(A ∧ B) = σ(A), σ(B) σ(B) = •σ(B) σ(^B) = ∗ • ∗σ(B)

(recall: Ci , Dj built from >, ∧, and ^)

Revantha Ramanayake (TU Wien) An introduction to the display calculus 43 / 66 Example

σ(>) = I

σ(p) = Xp σ(A ∧ B) = σ(A), σ(B) σ(B) = •σ(B) σ(^B) = ∗ • ∗σ(B)

Consider ^p → ^p. We compute the rule (∗ • ∗) • X ` Y ρ •(∗ • ∗)X ` Y By Kracht’s theorem

δKt + ρ is a display calculus for the tense logic Kt + ^p → ^p

Since ^p → ^p ↔ ^p → ^p

δK + ρ is a display calculus for the modal logic K + ^p → ^p

Revantha Ramanayake (TU Wien) An introduction to the display calculus 44 / 66 Another look at constructing display calculi

We have seen how to construct a display calculus for δKt. This raises several questions. How did we know which structural connectives to add? How to choose the display rules to ensure display property? Why did we consider Kt and not K ? Under what conditions can the program of adding structural connectives, display rules be used to obtain analytic calculi?

Revantha Ramanayake (TU Wien) An introduction to the display calculus 45 / 66 Residuation crucial to constructing new calculi

To add the tense operators we used the observation:

A → B ∈ Kt ⇔ A → B ∈ Kt

We then assigned the structural connective • to (, ).

Residuation is the key to constructing new display calculi.

Let us illustrate by constructing a calculus for bi-intuitionistic logic. . .

Revantha Ramanayake (TU Wien) An introduction to the display calculus 46 / 66 Bi-intuitionistic logic

Intuitionistic logic Ip ⊂ Cp and Ip + p ∨ ¬p = Cp. Aside: Gentzen observed that restricting the succedent of the sequent calculus SCp to at most one formula gives a sequent calculus SIp for Ip Ie. use sequents of the form X ` A and X ` instead of X ` Y (try to derive ` p ∨ ¬p in SIp and see what happens!) The language of bi-intuitionistic logic Bi−Ip extends the language of Ip with the connective →d (dual-implication). Axiomatisation for Bi−Ip were given by Rauszer, 1974.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 47 / 66 Residuated pairs for bi-intuitionistic logic

The following are theorems of Bi−Ip.

A → (B → C) ⇔ A ∧ B → C ⇔ B → (A → C)

B → (A ∨ C) ⇔ (A →d B) → C ⇔ A → (B ∨ C)

Assign the structural connective ◦ for (∧, →) and • for (→d , ∨). This immediately gives us the display rules: X ` Y ◦ Z X ` Y • Z X ◦ Y ` Z X • Y ` Z Y ` X ◦ Z Y ` X • Z and the following rewrite rules: A ◦ B ` Y X ` A ◦ B ∧l →r A ∧ B ` Y X ` A → B A • B ` Y X ` A • B →d l ∨r A →d B ` Y X ` A ∨ B

Revantha Ramanayake (TU Wien) An introduction to the display calculus 48 / 66 Computing the decoding rules

A ◦ B ` Y X ` A ◦ B ∧l →r A ∧ B ` Y X ` A → B A • B ` Y X ` A • B →d l ∨r A →d B ` Y X ` A ∨ B Here are the missing introduction rules (decoding rules in the terminology of Goré, 1998). X ` AY ` B X ` AB ` Y X ◦ Y ` A ∧ B A → B ` X ◦ Y X ` BA ` Y A ` XB ` Y X • Y ` B →d A A ∨ B ` X • Y Constructing the decoding rules is systematic (but not obvious, reasoning not shown here) and enforces:

Lemma Every rewrite rule is invertible.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 49 / 66 Some technical points

To be really precise, the semantics we used would actually lead to the non-associative Bi-Lambek (substructural) logic Ie. the ﬁrst residuation property is a ≤ (c ← b) ⇔ (a ⊗ b) ≤ c ⇔ b → (a → c) Since we were aiming for Bi−Ip we have used the properties of exchange, contraction, weakening and associativity...... to collapse

← and → ⊗ and ∧ ⊕ and ∨

The point is that we need to add structural rules for exchange, contraction, weakening and associativity!

Revantha Ramanayake (TU Wien) An introduction to the display calculus 50 / 66 Adding weakening, contraction, exchange, associativity

X ` Y X ` Y X ` Y • Z X ` Y • Z X ◦ Z ` Y X ` Z • Y X ◦ Z ` Y X ` Y • Y X ◦ X ` Y Z ◦ X ` Y X ` Y X ` Y X ` (Y • Z) • U (X ◦ Y ) ◦ Z ` U X ` Y • (Z • U) X ◦ (Y ◦ Z) ` U

Finally, the following structural rules are the unit rules for conjunction, disjunction. I ◦ X ` Y X ` Y • I X ` Y X ` Y X ◦ I ` Y X ` I • Y

Revantha Ramanayake (TU Wien) An introduction to the display calculus 51 / 66 A display calculus for Bi−Ip

A display sequent X ` Y is constructed from structures

Struc ::= Bi-Int formula | I | (X ◦ X) | (X • X)

I ` X X ` I >l ⊥r > ` X X ` ⊥ A ◦ B ` X X ` AX ` B ∧l ∧r A ∧ B ` X X ` A ∧ B A ` XB ` X X ` A • B ∨l ∨r A ∨ B ` X X ` A ∨ B X ` AY ` B X ` A ◦ B → l → r A → B ` X ◦ Y X ` A → B B • A ` X X ` BY ` A →d l →d r B →d A ` X X • Y ` B →d A

Revantha Ramanayake (TU Wien) An introduction to the display calculus 52 / 66 The display rules: Y ` X ◦ Z X ` Y ◦ Z X • Z ` Y X • Y ` Z X ◦ Y ` Z X ◦ Y ` Z X ` Y • Z X ` Y • Z X ` Y ◦ Z Y ` X ◦ Z X • Y ` Z X • Z ` Y Deﬁne the interpretation functions l and r from structures into Bi-Ip formulae:

l(A) = A r(A) = A l(I) = > r(I) = ⊥ l(X ◦ Y ) = l(X) ∧ l(Y ) r(X ◦ Y ) = l(X) → r(Y )

l(X • Y ) = l(X) →d l(Y ) r(X • Y ) = r(X) ∨ r(Y )

A sequent X ` Y is interpreted as l(X) → r(Y ).

Revantha Ramanayake (TU Wien) An introduction to the display calculus 53 / 66 Another look at constructing display calculi

Suitable gaggle semantics for a logic can be used to construct display calculi via the residuation property (Goré, 1998). Think non-associative Bi-Lambek calculus. The residuation property gives the display rules. Add new structural connectives and interpret as logical connectives (rewriting rules). Add remaining introduction rules (decoding rules). axioms for weakening, contraction etc. are converted to structural rules. This approach provides an answer to: which structural connectives to add? how to choose display rules?

Revantha Ramanayake (TU Wien) An introduction to the display calculus 54 / 66 Why did we consider Kt and not K ?

Consider the residuation property once more in Bi−Ip:

B → (A ∨ C) ⇔ (A →d B) → C ⇔ A → (B ∨ C)

Addition of the corresponding display rules leads to the addition of the dual-connective →d . As noted before: we can delete the introduction rules in δBi−Ip to get the display calculus δIp for Ip. Result follows by cut-elimination and conservativity of Bi−Ip over Ip. However, the translation of sequents in δIp is into Bi−Ip. In the same way, the property A → B ⇔ A → B (and the ensuing display rules) necessitate a detour into Kt.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 55 / 66 Structural rule extensions of display calculi: a general recipe

Logic L o / base display calculus C

suitable axiomatic extension structural rule extension L + {A1,..., An} o / C + ρ1 + ... + ρm

Generalises method for obtaining hypersequent structural rules from axioms (Ciabattoni et al., 2008) The approach is language and logic independent; purely syntactic conditions on the base calculus Extends Kracht’s theorem on primitive tense formulae.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 56 / 66 Obtaining a structural rule from a Hilbert axiom

δKt is a display calculus for the tense logic Kt satisfying C1–C8. Let us obtain the structural rule extension of δKt for the logic Kt ⊕ ^p → ^p. STEP 1. Start with the axiom (below left) and apply all possible invertible rules backwards (below right).

stop here: l, ^r not invertible ∗ • ∗p ` •^p ^l ^p ` •^p I ` ^p → ^p r ^p ` ^p →r I ` ^p → ^p

So it sufﬁces to introduce a structural rule equivalent to ∗ • ∗p ` •^p.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 57 / 66 STEP 2. Apply Ackermann’s Lemma.

Lemma The following rules are pairwise equivalent

S S A ` L S S L ` A ρ1 ρ2 δ δ X ` A X ` L A ` X 1 L ` X 2 where S is a set of sequents, L is a fresh schematic structure variable, and A is a tense formula.

d.p. lem L ` p ⇔ ⇔ ∗ • ∗p ` •^p p ` (∗ • ∗) • ^p L ` (∗ • ∗) • ^p

d.p. L ` p lem L ` p ^p ` M Stop when there are no ⇔ ⇔ more formulae in the •(∗ • ∗)L ` p •(∗ • ∗)L ` M ^ conclusion

Revantha Ramanayake (TU Wien) An introduction to the display calculus 58 / 66 STEP 3. Apply all possible invertible rules backwards.

L ` •p ∗ • ∗p ` M L ` p ^p ` M ⇔ L ` p p ` M •(∗ • ∗)L ` M ^ •(∗ • ∗)L ` M

The following rule is not a structural rule.

L ` •p ∗ • ∗p ` M ρ •(∗ • ∗)L ` M By Belnap’s general cut-elimination theorem, δKt + ρ has cut-elimination but not subformula property.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 59 / 66 STEP 4. Apply all possible cuts (and verify termination)

L ` •p ∗ • ∗p ` M d.p. •L ` p p ` ∗ • ∗M ρ ⇔ •(∗ • ∗)L ` M •(∗ • ∗)L ` M

•L ` ∗ • ∗M ⇔ ρ0 •(∗ • ∗)L ` M One direction is cut, the other direction is non-trivial. We conclude: δKt + ρ0 is a calculus for Kt + ^p → ^p with cut-elimination and subformula property.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 60 / 66 Summary of the recipe

(1) Invertible rules (2) Ackermann’s lemma (3) invertible rules (4) all possible cuts Only certain axioms are suitably decomposable Suppose we start with the axiom ^p → ^p.

(1) ⇔ I ` ^p → ^p ∗ • ∗p ` •^p

(2) L ` p ^p ` M (3) L ` •p ∗ • ∗p ` M ⇔ ⇔ ⇔?? •(∗ • ∗)L ` M •(∗ • ∗)L ` M From the display property, the last rule is equivalent to the following. Observe that we can no longer apply cut.

•L ` p p ` (∗ • ∗)M •(∗ • ∗)L ` M

Revantha Ramanayake (TU Wien) An introduction to the display calculus 61 / 66 Deﬁnition. Amenable calculus

Let C be a display calculus satisfying C1–C8. l and r are functions from structures into formulae s.t. l(A) = r(A) = A . Also: (i) X ` l(X) and r(X) ` X are derivable. (ii) X ` Y derivable implies l(X) ` r(Y ) is derivable. There is a structure constant I such that the following are admissible: I ` X X ` I Il Ir Y ` X X ` Y There are associative and commutative binary logical connectives ∨, ∧ in C such that

(a)∨ A ` X and B ` X implies ∨(A, B) ` X

(b)∨ X ` A implies X ` ∨(A, B) for any formula B.

(a)∧ X ` A and X ` B implies X ` ∧(A, B)

(b)∧ A ` X implies ∧(A, B) ` X for any formula B.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 62 / 66 The intermediate logic Ip + Bd2

Consider the intuitionistic axiom Bd2: p2 ∨ (p2 → (p1 ∨ ¬p1)). Here is the corresponding structural rule

M ` L K ` N ρ ` L • (M ◦ (N• (K ◦ I))) such that δIp + ρ is a cut-free calculus for Ip + Bd2.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 63 / 66 Recovering the display calculus δCp

Work out the structural rule ρ corresponding to p ∨ ¬p (ie. p ∨ p → ⊥). Observe that this is not exactly the calculus δCp we presented before. Fun exercise. Work out how to obtain δCp from δIp + ρ.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 64 / 66 Summary I

The display calculus generalises the sequent calculus by the addition of new structural connectives. Display rules yield the display property. The display property is used to prove Belnap’s general cut-elimination theorem. Residuation property central to choosing structural connectives, display rules. Relationship between cut-elimination and algebraic completions (recall Terui’s remark) Remember: the display calculus is one of several proof-frameworks proposed to address the (lack of) analytic sequent calculi for logics of interests. Some other frameworks include hypersequents, nested sequents, labelled sequents.

Revantha Ramanayake (TU Wien) An introduction to the display calculus 65 / 66 Summary II

In some frameworks such as the calculus of structures, we can operate ‘inside’ formulae (deep inference). The display calculus (below right) seems to mimic some notion of deep inference. I ` •B ` B •I ` B ` (B ∨ B0) •I ` B, B0 I ` •(B, B0) Structural rules in the display calculus means that it is difﬁcult to control proof-search (for example) However it is a good starting point for constructing an analytic calculus. Recent work used a display calculus as the startting point for an analytic calculus for Full intuitionistic linear logic (MILL extended with par). A (deep inference) nested sequent calculus is then constructed to obtain complexity, conservativity results (Clouston et al., 2013).

Revantha Ramanayake (TU Wien) An introduction to the display calculus 66 / 66 N.D. Belnap. Display Logic. Journal of Philosophical Logic, 11(4), 375–417, 1982. A. Ciabattoni, N. Galatos and K. Terui. From axioms to analytic rules in nonclassical logics. Proceedings of LICS 2008, pp. 229–240, 2008. A. Ciabattoni and R. Ramanayake. Structural rule extensions of display calculi: a general recipe. Proceedings of WOLLIC 2013. R. Clouston, R. Goré, and A. Tiu Annotation-Free Sequent Calculi for Full Intuitionistic Linear Logic. Proceedings of CSL 2013. G. Gentzen. The collected papers of Gerhard Gentzen. Edited by M. E. Szabo. Studies in Logic and the Foundations of Mathematics. Amsterdam, 1969. R. Goré. Substructural Logics On Display. Logic Journal of the IGPL, 6(3):451-504, 1998. R. Goré. Gaggles, Gentzen and Galois: how to display your favourite substructural logic. Logic Journal of the IGPL, 6(5):669-694, 1998. M. Kracht. Power and weakness of the modal display calculus. In: Proof theory of modal logic, 93–121. Kluwer. 1996. C. Rauszer. A formalization of propositional calculus of H-B logic. Studia Logica, 33. 1974. The slides can be found at

Revantha Ramanayake (TU Wien) An introduction to the display calculus 66 / 66